A note on hypergraph colorings

TL;DR

This paper introduces a sufficient condition for the existence of a specific type of hypergraph coloring called $(t,s)$-coloring, using the symmetric lopsided Lovász Local Lemma, generalizing previous results.

Contribution

It provides a new sufficient condition for $(t,s)$-colorings of hypergraphs, extending known results through probabilistic methods.

Findings

Established a sufficient condition for $(t,s)$-colorings

Generalized several existing hypergraph coloring results

Utilized the symmetric lopsided Lovász Local Lemma

Abstract

Let $t\geqslant 2$ and $s\geqslant 1$ be two integers. Define a $(t,s)$-coloring of a hypergraph to be a coloring of its vertices using $t$ colors such that each color appears on each edge at least $s$ times. In this note, we provide a sufficient condition for the existence of a $(t,s)$-coloring of a hypergraph by using the symmetric lopsided version of Lov\'asz Local Lemma. Our result generalizes several known results on hypergraph colorings.